Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 18496.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18496.e1 | 18496g4 | \([0, 1, 0, -2090433, -804591713]\) | \(159661140625/48275138\) | \(305461847512716935168\) | \([2]\) | \(663552\) | \(2.6358\) | |
| 18496.e2 | 18496g3 | \([0, 1, 0, -1905473, -1012893665]\) | \(120920208625/19652\) | \(124348401185718272\) | \([2]\) | \(331776\) | \(2.2892\) | |
| 18496.e3 | 18496g2 | \([0, 1, 0, -795713, 272874271]\) | \(8805624625/2312\) | \(14629223668908032\) | \([2]\) | \(221184\) | \(2.0865\) | |
| 18496.e4 | 18496g1 | \([0, 1, 0, -55873, 3128607]\) | \(3048625/1088\) | \(6884340550074368\) | \([2]\) | \(110592\) | \(1.7399\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18496.e have rank \(1\).
Complex multiplication
The elliptic curves in class 18496.e do not have complex multiplication.Modular form 18496.2.a.e
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
